Mathematical Sciences: New Directions in the Phase Field Approach to Free Boundary Problems

获得这一奖项后，首席研究员将继续使用渐近方法对自由边界问题进行分析研究，并将开始尝试构建有效的数值格式来解决界面现象。总体而言，他计划在他自己帮助推广的相场模型的基础上，发展出一个关于相变中出现的自由边界问题的统一理论。首席研究人员将考虑逐步逼真的合金模型以及与外部容器的接触。许多具有重要技术意义的自然现象涉及不同介质沿它们之间的界面或接触面的行为。有了这一奖项，首席研究员将继续他的重要工作，利用渐近方法开发相变模型的近似解，并将开始开发数值格式，以解决日益复杂的界面现象。这种渐近-数值相结合的方法应该对理解过渡层和自由边界问题的物理学有很大的好处。

With this award the principal investigator will continue his analytical studies of free boundary problems using asymptotic methods and he will initiate attempts to construct efficient numerical schemes for the resolution of interfacial phenomena. Overall he plans to develop a unified theory of free boundary problems that arise in phase transitions based upon the Phase Field Model that he himself has helped to popularize. The principal investigator will consider progressively realistic models of alloys and contact with external containers. Many technologically important natural phenomena involve the behavior of different media along the interfaces or surfaces of contact between them. With this award the principal investigator will continue his important work on developing approximate solutions of models of phase transitions using asymptotic methods and he will begin developing numerical schemes in order to resolve increasingly complicated interfacial phenomena. This combined asymptotic-numerical approach should pay big dividends toward understanding the physics of transition layers and free boundary problems.